ANSWER
The correct option is D.
EXPLANATION
If the parabola touches but does not cross the x-axis at
\(x = - 6\)
Then it means the root
\(x = - 6\)
repeats itself 2 times.
In order words the root
\(x = - 6\)
has a multiplicity of 2.
We can therefore write the equation
\({(x + 6})^{2} = 0\)
\((x + 6)(x + 6) = 0\)
We expand these to obtain
\({x}^{2} + 6x + 6x + 36 = 0\)
This implies that
\({x}^{2} + 12x + 36 = 0 - - - (1)\)
When we multiply through equation one by
\(- 1\)
we obtain,
\(- {x}^{2} - 12x - 36 - - (2)\)
The functions that corresponds to equation (1) and (2) are:
\(f(x) = {x}^{2} + 12x + 36\)
and
\(f(x) = - {x}^{2} - 12x - 36\)
respectively.
The above two parabolas have root
\(x = - 6\)
that does not cross the x-axis
